Redshifted formaldehyde from the gravitational lens B0218+357

The gravitational lens toward B0218+357 offers the unique possibility to study cool moderately dense gas with high sensitivity and angular resolution in a cloud that existed half a Hubble time ago. Observations of the radio continuum and six formaldehyde (H2CO) lines were carried out with the VLA, the Plateau de Bure interferometer, and the Effelsberg 100-m telescope. Three radio continuum maps indicate a flux density ratio between the two main images, A and B, of ~ 3.4 +/- 0.2. Within the errors the ratio is the same at 8.6, 14.1, and 43 GHz. The 1_{01}-0_{00} line of para-H2CO is shown to absorb the continuum of image A. Large Velocity Gradient radiative transfer calculations are performed to reproduce the optical depths of the observed two cm-wave K-doublet and four mm-wave rotational lines. These calculations also account for a likely frequency-dependent continuum cloud coverage. Confirming the diffuse nature of the cloud, an n(H2) density of < 1000 cm^{-3} is derived, with the best fit suggesting n(H2) ~ 200 cm^{-3}. The H2CO column density of the main velocity component is ~5 * 10^{13} cm^{-2}, to which about 7.5 * 10^{12} cm^{-2} has to be added to also account for a weaker feature on the blue side, 13 km/s apart. N(H2CO)/N(NH3) ~ 0.6, which is four times less than the average ratio obtained from a small number of local diffuse (galactic) clouds seen in absorption. The ortho-to-para H2CO abundance ratio is 2.0 - 3.0, which is consistent with the kinetic temperature of the molecular gas associated with the lens of B0218+357. With the gas kinetic temperature and density known, it is found that optically thin transitions of CS, HCN, HNC, HCO+, and N2H+ (but not CO) will provide excellent probes of the cosmic microwave background at redshift z=0.68.